Disclaimer: I lay no claim to the originality of the basic ideas expounded here. Its distinctness lies more in its scope and in the detail of its exposition.
As in any field of study, a particular system has an associated state. In quantum mechanics, the state of a system of particles over a space at a particular time is best described by what is known as the wave function. Conventionally written as,
The content of the
The jargon "wave function" is not an ordinary kind of function we met in calculus. Though it has some properties we've already encountered. For instance, a wave function has to be differentiable or smooth. In addition a wave function is a complex function, and by that I mean that it is not a value we can associate to the physical world. We have to do something about this wave function in order to obtain a value we can put into some kind of experiment.
Note, however, that by knowing the wave function, we can make some precise predictions of the probability of the various possible outcome of any given type of physical measurement on the particle. In particular, in a position measurement, the probability of finding the particle within some volume
Here we introduced the complex conjugate of the wave function
In some other standard, the equation above is written as,
If you're familiar with probability theory, it is easy to see that the function
We can also say that the wave function
If we take the integral of the equation we have from in all space, we know that we must find the particle somewhere hence a probability of one,
If the wave function has this property, it is said to be normalized. By normalized, we mean that total summation or integration must be equal to unity. If we don't have a normalized wave, we obtain an integration that is not unity. Note, however that we can normalized all wave function as long as it is integrable.
Lets say we have a system of state of definite energy E, (by definite we mean a non-fluctuating E). And lets assume we can express our wave function as a product of separate variables, that is:
Another way to express the time-dependence of the wave function is given by,
Note that this is always the form of the wave function's time dependence if we have a system of definite energy E.
How about a system of definite momentum?
If we have a system of definite momentum, the space dependent part
where
The
The last equation can be visualized as a moving wave. Or in other words, a wave function is actually a wave front that moves along the direction of the momentum p.
